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其它</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2021-12-06T09:36:00.000Z" title="发表于 2021-12-06 17:36:00">2021-12-06</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2021-12-10T12:59:46.902Z" title="更新于 2021-12-10 20:59:46">2021-12-10</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/hexo3/categories/%E7%AE%97%E6%B3%95%E5%9F%BA%E7%A1%80/">算法基础</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-wordcount"><i class="far fa-file-word fa-fw post-meta-icon"></i><span class="post-meta-label">字数总计:</span><span class="word-count">793</span><span class="post-meta-separator">|</span><i class="far fa-clock fa-fw post-meta-icon"></i><span class="post-meta-label">阅读时长:</span><span>3分钟</span></span><span class="post-meta-separator">|</span><span class="post-meta-pv-cv" id="" data-flag-title="算法 - 其它"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">阅读量:</span><span id="busuanzi_value_page_pv"></span></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="算法-其它"><a href="#算法-其它" class="headerlink" title="算法 - 其它"></a>算法 - 其它</h1><h2 id="汉诺塔"><a href="#汉诺塔" class="headerlink" title="汉诺塔"></a>汉诺塔</h2><div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/69d6c38d-1dec-4f72-ae60-60dbc10e9d15.png" width="300"/> </div><br>

<p>有三个柱子，分别为 from、buffer、to。需要将 from 上的圆盘全部移动到 to 上，并且要保证小圆盘始终在大圆盘上。</p>
<p>这是一个经典的递归问题，分为三步求解：</p>
<p>① 将 n-1 个圆盘从 from -&gt; buffer</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/f9240aa1-8d48-4959-b28a-7ca45c3e4d91.png" width="300"/> </div><br>

<p>② 将 1 个圆盘从 from -&gt; to</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/f579cab0-3d49-4d00-8e14-e9e1669d0f9f.png" width="300"/> </div><br>

<p>③ 将 n-1 个圆盘从 buffer -&gt; to</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/d02f74dd-8e33-4f3c-bf29-53203a06695a.png" width="300"/> </div><br>

<p>如果只有一个圆盘，那么只需要进行一次移动操作。</p>
<p>从上面的讨论可以知道，a<sub>n</sub> = 2 * a<sub>n-1</sub> + 1，显然 a<sub>n</sub> = 2<sup>n</sup> - 1，n 个圆盘需要移动 2<sup>n</sup> - 1 次。</p>
<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Hanoi</span> </span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">move</span><span class="params">(<span class="keyword">int</span> n, String from, String buffer, String to)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">if</span> (n == <span class="number">1</span>) &#123;</span><br><span class="line">            System.out.println(<span class="string">&quot;from &quot;</span> + from + <span class="string">&quot; to &quot;</span> + to);</span><br><span class="line">            <span class="keyword">return</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        move(n - <span class="number">1</span>, from, to, buffer);</span><br><span class="line">        move(<span class="number">1</span>, from, buffer, to);</span><br><span class="line">        move(n - <span class="number">1</span>, buffer, from, to);</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>&#123;</span><br><span class="line">        Hanoi.move(<span class="number">3</span>, <span class="string">&quot;H1&quot;</span>, <span class="string">&quot;H2&quot;</span>, <span class="string">&quot;H3&quot;</span>);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<figure class="highlight html"><table><tr><td class="code"><pre><span class="line">from H1 to H3</span><br><span class="line">from H1 to H2</span><br><span class="line">from H3 to H2</span><br><span class="line">from H1 to H3</span><br><span class="line">from H2 to H1</span><br><span class="line">from H2 to H3</span><br><span class="line">from H1 to H3</span><br></pre></td></tr></table></figure>

<h2 id="哈夫曼编码"><a href="#哈夫曼编码" class="headerlink" title="哈夫曼编码"></a>哈夫曼编码</h2><p>根据数据出现的频率对数据进行编码，从而压缩原始数据。</p>
<p>例如对于一个文本文件，其中各种字符出现的次数如下：</p>
<ul>
<li>a : 10</li>
<li>b : 20</li>
<li>c : 40</li>
<li>d : 80</li>
</ul>
<p>可以将每种字符转换成二进制编码，例如将 a 转换为 00，b 转换为 01，c 转换为 10，d 转换为 11。这是最简单的一种编码方式，没有考虑各个字符的权值（出现频率）。而哈夫曼编码采用了贪心策略，使出现频率最高的字符的编码最短，从而保证整体的编码长度最短。</p>
<p>首先生成一颗哈夫曼树，每次生成过程中选取频率最少的两个节点，生成一个新节点作为它们的父节点，并且新节点的频率为两个节点的和。选取频率最少的原因是，生成过程使得先选取的节点位于树的更低层，那么需要的编码长度更长，频率更少可以使得总编码长度更少。</p>
<p>生成编码时，从根节点出发，向左遍历则添加二进制位 0，向右则添加二进制位 1，直到遍历到叶子节点，叶子节点代表的字符的编码就是这个路径编码。</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/8edc5164-810b-4cc5-bda8-2a2c98556377.jpg" width="300"/> </div><br>

<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Huffman</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">private</span> <span class="class"><span class="keyword">class</span> <span class="title">Node</span> <span class="keyword">implements</span> <span class="title">Comparable</span>&lt;<span class="title">Node</span>&gt; </span>&#123;</span><br><span class="line">        <span class="keyword">char</span> ch;</span><br><span class="line">        <span class="keyword">int</span> freq;</span><br><span class="line">        <span class="keyword">boolean</span> isLeaf;</span><br><span class="line">        Node left, right;</span><br><span class="line"></span><br><span class="line">        <span class="function"><span class="keyword">public</span> <span class="title">Node</span><span class="params">(<span class="keyword">char</span> ch, <span class="keyword">int</span> freq)</span> </span>&#123;</span><br><span class="line">            <span class="keyword">this</span>.ch = ch;</span><br><span class="line">            <span class="keyword">this</span>.freq = freq;</span><br><span class="line">            isLeaf = <span class="keyword">true</span>;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="function"><span class="keyword">public</span> <span class="title">Node</span><span class="params">(Node left, Node right, <span class="keyword">int</span> freq)</span> </span>&#123;</span><br><span class="line">            <span class="keyword">this</span>.left = left;</span><br><span class="line">            <span class="keyword">this</span>.right = right;</span><br><span class="line">            <span class="keyword">this</span>.freq = freq;</span><br><span class="line">            isLeaf = <span class="keyword">false</span>;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="meta">@Override</span></span><br><span class="line">        <span class="function"><span class="keyword">public</span> <span class="keyword">int</span> <span class="title">compareTo</span><span class="params">(Node o)</span> </span>&#123;</span><br><span class="line">            <span class="keyword">return</span> <span class="keyword">this</span>.freq - o.freq;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> Map&lt;Character, String&gt; <span class="title">encode</span><span class="params">(Map&lt;Character, Integer&gt; frequencyForChar)</span> </span>&#123;</span><br><span class="line">        PriorityQueue&lt;Node&gt; priorityQueue = <span class="keyword">new</span> PriorityQueue&lt;&gt;();</span><br><span class="line">        <span class="keyword">for</span> (Character c : frequencyForChar.keySet()) &#123;</span><br><span class="line">            priorityQueue.add(<span class="keyword">new</span> Node(c, frequencyForChar.get(c)));</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">while</span> (priorityQueue.size() != <span class="number">1</span>) &#123;</span><br><span class="line">            Node node1 = priorityQueue.poll();</span><br><span class="line">            Node node2 = priorityQueue.poll();</span><br><span class="line">            priorityQueue.add(<span class="keyword">new</span> Node(node1, node2, node1.freq + node2.freq));</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> encode(priorityQueue.poll());</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">private</span> Map&lt;Character, String&gt; <span class="title">encode</span><span class="params">(Node root)</span> </span>&#123;</span><br><span class="line">        Map&lt;Character, String&gt; encodingForChar = <span class="keyword">new</span> HashMap&lt;&gt;();</span><br><span class="line">        encode(root, <span class="string">&quot;&quot;</span>, encodingForChar);</span><br><span class="line">        <span class="keyword">return</span> encodingForChar;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">void</span> <span class="title">encode</span><span class="params">(Node node, String encoding, Map&lt;Character, String&gt; encodingForChar)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">if</span> (node.isLeaf) &#123;</span><br><span class="line">            encodingForChar.put(node.ch, encoding);</span><br><span class="line">            <span class="keyword">return</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        encode(node.left, encoding + <span class="string">&#x27;0&#x27;</span>, encodingForChar);</span><br><span class="line">        encode(node.right, encoding + <span class="string">&#x27;1&#x27;</span>, encodingForChar);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">文章作者: </span><span class="post-copyright-info"><a href="mailto:undefined">Zhang Shuo</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">文章链接: </span><span class="post-copyright-info"><a href="https://zhang-shuo-fr.gitee.io/hexo3/2021/12/06/notes/%E7%AE%97%E6%B3%95%20-%20%E5%85%B6%E5%AE%83/">https://zhang-shuo-fr.gitee.io/hexo3/2021/12/06/notes/%E7%AE%97%E6%B3%95%20-%20%E5%85%B6%E5%AE%83/</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">版权声明: </span><span class="post-copyright-info">本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank">CC BY-NC-SA 4.0</a> 许可协议。转载请注明来自 <a href="https://zhang-shuo-fr.gitee.io/hexo3" target="_blank">Zhang Shuo'blog</a>！</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/hexo3/tags/%E7%AE%97%E6%B3%95%E5%9F%BA%E7%A1%80/">算法基础</a></div><div class="post_share"><div class="social-share" data-image="/hexo3/img/16.jpg" data-sites="facebook,twitter,wechat,weibo,qq"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/social-share.js/dist/css/share.min.css" media="print" onload="this.media='all'"><script src="https://cdn.jsdelivr.net/npm/social-share.js/dist/js/social-share.min.js" defer></script></div></div><div class="post-reward"><div class="reward-button button--animated"><i class="fas fa-qrcode"></i> 打赏</div><div class="reward-main"><ul class="reward-all"><li class="reward-item"><a href="/hexo3/img/wechat.jpg" target="_blank"><img class="post-qr-code-img" src= "" data-lazy-src="/hexo3/img/wechat.jpg" alt="微信"/></a><div class="post-qr-code-desc">微信</div></li><li class="reward-item"><a href="/hexo3/img/alipay.jpg" target="_blank"><img class="post-qr-code-img" src= "" data-lazy-src="/hexo3/img/alipay.jpg" alt="支付宝"/></a><div class="post-qr-code-desc">支付宝</div></li></ul></div></div><nav class="pagination-post" id="pagination"><div class="prev-post pull-left"><a href="/hexo3/2021/12/06/notes/%E7%AE%97%E6%B3%95%20-%20%E5%B9%B6%E6%9F%A5%E9%9B%86/"><img class="prev-cover" src= "" data-lazy-src="/hexo3/img/19.jpg" onerror="onerror=null;src='/hexo3/img/404.jpg'" alt="cover of previous post"><div class="pagination-info"><div class="label">上一篇</div><div class="prev_info">算法 - 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